%
% macros for differentials, partial derivatives, Jacobian matrices
\def\vectorsmal#1{#1}
\def\vecsmal#1{\vbox{\ialign{##\crcr
    \rightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
    $\hfil\scriptstyle{#1}\hfil$\crcr}}}
\def\monom#1#2#3{#1_{1}^{#2_{1}}\cdots #1_{#3}^{#2_{#3}}}
\def\Sym#1{\hbox{Sym}^#1}
\def\Ispac#1#2{#1^{(#2)}}
%\def\affspac#1#2{{\bf A}^{#1}_{#2}}
\def\affspac#1#2{{\mathbb A}^{#1}_{#2}}
%\def\affreal{{\bf A}}
\def\affreal{{\mathbb A}}
\def\fixset{\mathrm{Fix}}
\def\mod{\hbox{mod}}
\def\sgnam{\hbox{sg}}
\def\volum{\hbox{vol}}
\def\Lambatp#1{\Lambda_{\mathrm{at}\>#1}}
\def\Rat#1#2{\mathrm{Rat}(#1,#2)}
\def\Mer#1{\s{M}\mathrm{er}(#1)}
\def\Irred#1{\mathrm{Irred}/#1}
\def\FncFlds#1{\mathrm{FncFlds}/#1}
\def\trace{\mathrm{tr}}
\def\Sing{\mathrm{Sing}}
\def\qtrace{\hbox{Tr}}
\def\diag{\hbox{diag}}
\def\Gram{\hbox{Gram}}
\def\Ad{\mathrm{Ad}}
\def\Frac{\mathrm{Frac}}
\def\app{\hbox{app}}
\def\smad{\mathrm{ad}}
%\def\eucreal{{\bf E}}
\def\eucreal{{\mathbb E}}
\def\qone{{\bf 1}}
\def\qi{{\bf i}}
\def\qj{{\bf j}}
\def\qk{{\bf k}}
\def\quat{{\mathbb H}}
\def\setscat{\mathbf{Sets}}
\def\grcat{\mathbf{Grp}}
\def\abcat{\mathbf{Ab}}
\def\crngcat{\mathbf{CRng}}
\def\kalgcat{\hbox{\bf k-alg}}
\def\affalgcat{\hbox{\bf k-affalg}}
\def\Ob{\mathrm{Ob}}
\def\Mor{\mathrm{Mor}}
\def\Ar{\mathrm{Ar}}
%
\def\res{|}
\def\brokrarrow{\>\raise1.5pt\hbox{$\scriptstyle -\, -\, \rightarrow$}\>}
\def\ratmap#1#2#3{#1\co #2 \brokrarrow #3}
%\def\ratmap#1#2#3{#1\co #2 - -\rightarrow #3}
\def\blowup#1#2{\hbox{Bl}_{#1} #2}
\def\mGL#1{{\bf GL}(#1)}
\def\mGA#1{{\bf GA}(#1)}
\def\mGAa#1{{\bf GA}_a(#1)}
\def\cech#1{\check{#1}}
\def\mE#1{{\bf E}(#1)}
\def\mIs#1{{\bf Is}(#1)}
\def\mMo#1{{\bf Mo}(#1)}
\def\mPGL#1{{\bf PGL}(#1)}
\def\mSL#1{{\bf SL}(#1)}
\def\mEA#1{{\bf EA}(#1)}
\def\mIA#1{{\bf IA}(#1)}
\def\mSE#1{{\bf SE}(#1)}
\def\mU#1{{\bf U}(#1)}
\def\mUpq#1#2{{\bf U}(#1,#2)}
\def\mSU#1{{\bf SU}(#1)}
\def\mSUpq#1#2{{\bf SU}(#1,#2)}
\def\mSA#1{{\bf SA}(#1)}
\def\mPSL#1{{\bf PSL}(#1)}
\def\mO#1{{\bf O}(#1)}
\def\mOpq#1#2{{\bf O}(#1,#2)}
\def\mSO#1{{\bf SO}(#1)}
\def\mSOpq#1#2{{\bf SO}(#1,#2)}
\def\mPSO#1{{\bf PSO}(#1)}
\def\mAut#1{{\bf Aut}(#1)}
\def\mDIL#1{{\bf DIL}(#1)}
\def\Grassm{{\bf G}}
\def\PGrassm{{\bf PG}}
\def\zozo#1{\vector{#1}}
\def\zozor#1{\novect{#1}}
\def\affvec#1{\vector{#1}}
\def\intd{\int\!\!\!\int}
\def\hli#1{\widehat{#1}}
\def\osc#1#2{{\rm Osc}_{#1} #2}
\def\Span{{\rm Span\/}}
\def\fallpow#1#2{#1^{\underline{#2}}}
\def\ringpoly#1#2{#1[#2]}
\def\fracfpoly#1#2{#1(#2)}
\def\ftranspos#1{\hbox{$^{t}#1$}}
\def\transpos#1{#1^{\top}}
\def\flecheabov#1{\buildrel #1\over\longrightarrow}
\def\flechea#1#2#3{#1\,\flecheabov{#2}\,#3}
\def\flecheb#1#2#3#4#5{#1\,\flecheabov{#2}\,#3\,\flecheabov{#4}\,#5}
\def\flechec#1#2#3#4#5#6#7{#1\,\flecheabov{#2}\,#3
\,\flecheabov{#4}\,#5\,\flecheabov{#6}\,#7}
\def\fleched#1#2#3#4#5#6#7#8#9{#1\,\flecheabov{#2}\,#3\,\flecheabov{#4}\,#5
\,\flecheabov{#6}\,#7\,\flecheabov{#8}\,#9}
\def\flechel#1#2{#1\,\flecheabov{#2}\,}
\def\flecher#1#2{\,\flecheabov{#1}\,#2}
\def\shortexact#1#2#3#4#5{0\,\flecheabov{}\,#1\,\flecheabov{#2}\,#3
\,\flecheabov{#4}\,#5\,\flecheabov{}\,0}
\def\libvec#1#2{\overrightarrow{#1#2}}
\def\toto#1#2{\overrightarrow{#1#2}}
\def\libvecb#1#2{#2 - #1}
\def\libvecbo#1#2{{\bf #1#2}}
%\def\vectorr#1{\overrightarrow{#1}}
\def\vectorr#1{#1}
\def\bvect#1{{\bf #1}}
\def\novect#1{#1}
\def\vector#1{\fakoverrightarrow{#1}}
\def\ptb#1{\overline{#1}}
\def\covec#1{#1^{*}}
\def\ortho#1{#1^{0}}
\def\orthog#1{#1^{\perp}}
\def\biortho#1{#1^{00}}
\def\biorthog#1{#1^{\perp\perp}}
\def\translat#1#2{#1 + \novect{#2}}
\def\translatv#1#2{#1 + \vector{#2}}
\def\binvec#1#2{\pmatrix{#1\cr #2\cr}}
\def\trivec#1#2#3{\pmatrix{#1\cr #2\cr #3\cr}}
\def\bincoef#1#2{#1\choose #2}
\def\mapdef#1#2#3{#1\co #2\rightarrow #3}
\def\famil#1#2#3{(#1_{#2})_{#2\in #3}}
\def\familvec#1#2#3{(\vector{#1_{#2}})_{#2\in #3}}
\def\linspac#1#2{{\rm L}(#1; #2)}
\def\slinspac#1#2{{\rm S}(#1; #2)}
\def\clinspac#1#2#3{{\cal L}_{#1}(#2; #3)}
\def\lincomb#1#2#3#4{\sum_{#3\in #4} #1_{#3}\vector{#2_{#3}}}
\def\lincombin#1#2#3#4{\sum_{#3\in #4} #1_{#3} #2_{#3}}
\def\linsum#1#2#3{#1_{1}\vector{#2_{1}} + \cdots + #1_{#3}\vector{#2_{#3}}} 
\def\linsom#1#2#3{#1_{1}#2_{1} + \cdots + #1_{#3}#2_{#3}} 
\def\pmatrice#1{\left( \matrice{#1} \right)}
\def\dmatrice#1{\left| \matrix{#1} \right|}
%\def\dmatrice#1{\left\mid \matrix{#1} \right\mid}
\def\dmatriceb#1{\left| \matrice{#1} \right|}
\def\colvec#1#2{\pmatrix{#1_{1}\cr \vdots\cr #1_{#2}\cr}}
\def\trivec#1#2#3{\pmatrix{#1\cr #2\cr #3\cr}}
\def\colvector#1#2{\pmatrix{#1_{1}\cr #1_{2}\cr\vdots\cr #1_{#2}\cr}}
\def\linvec#1#2{(#1_{1},\ldots, #1_{#2})}
\def\specrow#1#2#3{(\matel{#1}{#2}{1},\ldots,\matel{#1}{#2}{#3})}
\def\speccol#1#2#3{\pmatrix{\matel{#1}{1}{#2}\cr \vdots\cr 
\matel{#1}{#3}{#2}\cr}}
\def\linbasis#1#2{(\vector{#1_{1}},\ldots, \vector{#1_{#2}})} 
\def\lbasis#1#2{(#1_{1},\ldots, #1_{#2})} 
\def\eunorme#1#2{\left(|#1_{1}|^{2} + \cdots + |#1_{#2}|^{2}\right)^{1\over 2}}
\def\eunorm#1#2{\sqrt{|#1_{1}|^{2} + \cdots + |#1_{#2}|^{2})}}
\def\eudist#1#2#3{\left(|#1_{1} - #2_{1}|^{2}+ \cdots + 
|#1_{#3} - #2_{#3}|^{2}\right)^{1\over 2}}
\def\norme#1{\left\|#1\right\|}
\def\smnorme#1{\|#1\|}
\def\dist#1#2#3{d_{#1}(#2,\,#3)}
\def\inprod#1#2{(#1 | #2)}
\def\dotprod#1#2{#1\cdot #2}
\def\absval#1#2{|#1 - #2|}
\def\cloball#1#2{B(#1,#2)}
\def\opball#1#2{B_{0}(#1,#2)}
\def\ncloball#1{B(#1)}
\def\nopball#1{B_{0}(#1)}
\def\adher#1{\overline{#1}}
\def\interio#1{\buildrel \circ\over #1}
\def\fr#1{{\rm Fr}\>#1}
%
\def\polynom#1#2#3{#1_{0}+#1_{1}#3+\cdots +#1_{#2}#3^{#2}}
\def\hpolynom#1#2#3{#1_{#2}#3^{#2} + #1_{#2 - 1}#3^{#2 - 1} + \cdots  + #1_{0}}
\def\derivpol#1#2#3{#2#1_{#2}#3^{#2 - 1} + (#2 - 1)#1_{#2 - 1}#3^{#2 - 2} 
+ \cdots + 2#1_{2}#3 + #1_{1}}
\def\rpolynom#1#2#3{#1_{0}#3^{#2} + #1_{1}#3^{#2 - 1} + \cdots  + #1_{#2}}
\def\monic#1#2#3{#3^{#2} + #1_{#2 - 1}#3^{#2 - 1} + \cdots  + #1_{0}}
\def\rmonic#1#2#3{#3^{#2} + #1_{1}#3^{#2 - 1} + \cdots  + #1_{#2}}
\def\pcoef#1#2#3{#1_{(#2_{1},\ldots,#2_{#3})}}
\def\mpolynom#1#2#3#4{\sum_{(#2_{1},\ldots,#2_{#3})\in\natnums^{(#3)}}
\pcoef{#1}{#2}{#3}\monom{#4}{#2}{#3}}
\def\myfrac#1#2{{\displaystyle {#1\strut\over\strut #2}}}
%\def\dfrac#1#2{{\displaystyle #1 \strut\over\strut \displaystyle #2}}
\def\dfrac#1#2{{\displaystyle {{\displaystyle #1}\strut\over\strut #2}}}
\def\batop#1#2{{\displaystyle #1\atop \displaystyle #2}}
\def\derivpderivDo#1{#1'}
\def\derivp#1{#1'}
\def\derivD#1#2{{\rm D}^{#2}#1}
\def\derivDof#1#2{{\rm D}#1#2}
\def\vfderiv#1#2{{\rm D}_{#1}#2}
\def\parderiv#1#2{\displaystyle{{\partial{#1}}\strut\over\strut{\partial{#2}}}}
\def\parderivb#1#2{\displaystyle{{\partial^2{#1}}
\over\strut{\partial{#2^{2}}}}}
\def\parderivc#1#2#3{\displaystyle{{\partial^2{#1}}
\strut\over\strut{\partial{#2}\partial{#3}}}}
\def\parderivd#1#2#3#4#5{\displaystyle{{\partial^{#2}{#1}}
\strut\over\strut{\partial{#3_{#4}}\ldots\partial{#3_{#5}}}}}
\def\vparderiv#1#2{\displaystyle{{\partial{#1}}
\strut\over\strut{\partial{\vector{#2}}}}}
%
\def\vparderivh#1#2#3{\displaystyle{{\partial^{#3} {#1}}
\strut\over\strut{\partial{\vector{#2}}^{#3}}}}
%
%
\def\vparderivb#1#2#3{\displaystyle{%
\partial^{2} {#1}\strut\over\strut{%
\partial{\vector{#2}}\partial{\vector{#3}}%
}%
}%
}
%
\def\vparderivc#1#2#3#4#5{\displaystyle{%
\partial^{#4 + #5} {#1}\strut\over\strut{%
\partial{\vector{#2}}^{#4}\partial{\vector{#3}}^{#5}%
}%
}%
}
%
\def\Der#1#2#3#4{\mathrm{Der}_{#1}(#2, #3; #4)}
\def\sDer#1#2#3{\mathrm{Der}_{#1}(#2, #3)}
\def\Dparderiv#1#2{\partial_{#2}{#1}}
\def\Dparder#1#2{{\rm D}_{#2}{#1}}
\def\deriv#1#2{{\displaystyle{d#1\strut\over\strut d#2}}}
\def\dotderiv#1{{\dot #1}}
\def\dotdderiv#1{{\ddot #1}}
\def\vderiv#1#2{{\rm D}_{\vectorsmal{#1}}{#2}}
%
\def\vderivb#1#2#3{{\rm D}^{2}_{\vectorsmal{#1},\vectorsmal{#2}}{#3}}
\def\vderivc#1#2#3{{\rm D}_{\vectorsmal{#1}}{\rm D}_{\vectorsmal{#2}}{#3}}
\def\vderivd#1#2#3{{\rm D}_{\vectorsmal{#1}}\ldots
{\rm D}_{\vectorsmal{#2}}{#3}}
%
\def\vderivdd#1#2#3#4#5{\underbrace{{\rm D}_{\vectorsmal{#1}}\ldots
{\rm D}_{\vectorsmal{#2}}}_{i}\underbrace{{\rm D}_{\vectorsmal{#3}}\ldots
{\rm D}_{\vectorsmal{#4}}}_{j}\,{#5}}
%
\def\matdef#1#2{{\rm M}_{#1}#2}
\def\matdefin#1#2#3{{\rm M}_{#1, #2}#3}
%
\def\jacob#1#2#3#4#5{%
\pmatrix{\parderiv{#1_{1}}{#2_{1}}(#3)&\parderiv{#1_{1}}{#2_{2}}(#3)&\ldots&
\parderiv{#1_{1}}{#2_{#5}}(#3)\cr
\parderiv{#1_{2}}{#2_{1}}(#3)&\parderiv{#1_{2}}{#2_{2}}(#3)&\ldots&
\parderiv{#1_{2}}{#2_{#5}}(#3)\cr
\vdots&\vdots&\ddots&\vdots\cr
\parderiv{#1_{#4}}{#2_{1}}(#3)&\parderiv{#1_{#4}}{#2_{2}}(#3)&\ldots&
\parderiv{#1_{#4}}{#2_{#5}}(#3)\cr}}
%
%
\def\Djacob#1#2#3#4{%
\pmatrix{\Dparderiv{#1_{1}}{1}(#4)&\Dparderiv{#1_{1}}{2}(#4)&\ldots& 
\Dparderiv{#1_{1}}{#3}(#4)\cr
\Dparderiv{#1_{2}}{1}(#4)&\Dparderiv{#1_{2}}{2}(#4)&\ldots& 
\Dparderiv{#1_{2}}{#3}(#4)\cr
\vdots&\vdots&\ddots&\vdots\cr
\Dparderiv{#1_{#2}}{1}(#4)&\Dparderiv{#1_{#2}}{2}(#4)&\ldots& 
\Dparderiv{#1_{#2}}{#3}(#4)\cr}}
%
%
\def\cjacob#1#2#3#4#5{%
\pmatrix{\parderiv{#1}{#4}(#5)\cr
\parderiv{#2}{#4}(#5)\cr
\parderiv{#3}{#4}(#5)\cr}}
%
%
\def\ljacob#1#2#3#4#5{%
\bigg(\parderiv{#1}{#2}(#5)\>\> \parderiv{#1}{#3}(#5)\>\> 
\parderiv{#1}{#4}(#5)\bigg)}
%
%
\def\jacoba#1#2#3#4#5{%
\pmatrix{\parderiv{#1}{#3}(#5)&\parderiv{#1}{#4}(#5)\cr
\parderiv{#2}{#3}(#5)&\parderiv{#2}{#4}(#5)\cr}}
%
%
\def\jacobb#1#2#3#4#5#6{%
\pmatrix{\parderiv{#1}{#4}(#6)&\parderiv{#1}{#5}(#6)\cr
\parderiv{#2}{#4}(#6)&\parderiv{#2}{#5}(#6)\cr
\parderiv{#3}{#4}(#6)&\parderiv{#3}{#5}(#6)\cr}}
%
\def\rowjacob#1#2#3#4{%
\bigg(\parderiv{#1}{#2_{1}}(#3),\ldots, \parderiv{#1}{#2_{#4}}(#3)\bigg)}
%
\def\hessien#1#2#3#4{%
\pmatrix{\parderivb{#1}{#2_{1}}(#3)&\parderivc{#1}{#2_{1}}{#2_{2}}(#3)&\ldots&
\parderivc{#1}{#2_{1}}{#2_{#4}}(#3)\cr
\parderivc{#1}{#2_{1}}{#2_{2}}(#3)&\parderivb{#1}{#2_{2}}(#3)&\ldots&
\parderivc{#1}{#2_{2}}{#2_{#4}}(#3)\cr
\vdots&\vdots&\ddots&\vdots\cr
\parderivc{#1}{#2_{1}}{#2_{#4}}(#3)&\parderivc{#1}{#2_{2}}{#2_{#4}}(#3)&\ldots&
\parderivb{#1}{#2_{#4}}(#3)\cr}}
%
%
\def\matel#1#2#3{#1_{#2\, #3}}
%
%\def\mata#1#2#3#4{%
%\pmatrice{#1& #2\cr
%#3&#4\cr}}

\def\mata#1#2#3#4{%
\pmatrice{#1& #2\cr
#3&#4\cr}}

%
\def\matta#1#2#3#4{%
\left(
\matrix{
#1 & #2\cr
#3 & #4\cr
}
\right)}
%
%
\def\matb#1{%
\pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}& \matel{#1}{1}{3}\cr
\matel{#1}{2}{1}& \matel{#1}{2}{2}& \matel{#1}{2}{3}\cr
\matel{#1}{3}{1}& \matel{#1}{3}{2}& \matel{#1}{3}{3}\cr}}
%
%
\def\matc#1#2#3#4#5#6#7#8#9{%
\pmatrice{#1& #2& #3\cr
#4 & #5 & #6\cr
#7 & #8 & #9\cr}}
%
\def\mattc#1#2#3#4#5#6#7#8#9{%
\left(
\matrix{
#1 & #2 & #3\cr
#4 & #5 & #6\cr
#7 & #8 & #9\cr
}
\right)}
%

%
\def\genmat#1#2#3{%
\pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#3}\cr
\matel{#1}{2}{1}& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#3}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#3}\cr}}
%
\def\genmatrix#1#2#3{%
\pmatrix{\matel{#1}{1}{1}& \ldots& \matel{#1}{1}{#3}\cr
\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1}& \ldots& \matel{#1}{#2}{#3}\cr}}
%
\def\idmat{%
\pmatrix{\matel{1}{}{}& \matel{0}{}{}&\ldots& \matel{0}{}{}\cr
\matel{0}{}{}& \matel{1}{}{}&\ldots& \matel{0}{}{}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{0}{}{}& \matel{0}{}{}&\ldots& \matel{1}{}{}\cr}}
%
%
\def\uptmat#1#2#3{%
\pmatrix{
\matel{#1}{1}{1}&\matel{#1}{1}{2}&\matel{#1}{1}{3}&\ldots& 
\matel{#1}{1}{#3 -1}&\matel{#1}{1}{#3}\cr
0& \matel{#1}{2}{2}&\matel{#1}{2}{3}&\ldots& \matel{#1}{2}{#3 -1}& \matel{#1}{2}{#3}\cr
0& 0&\matel{#1}{3}{3}&\ldots& \matel{#1}{3}{#3 - 1}& \matel{#1}{3}{#3}\cr
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr
0& 0&0&\ldots&\matel{#1}{#2 -1}{#3 -1} &\matel{#1}{#2 - 1}{#3}\cr
0& 0&0&\ldots& 0 &\matel{#1}{#2}{#3}\cr}}
%
%
\def\uptmatb#1#2#3#4{%
\pmatrix{
#4_{1} - #4&\matel{#1}{1}{2}&\matel{#1}{1}{3}&\ldots& 
\matel{#1}{1}{#3 -1}&\matel{#1}{1}{#3}\cr
0& #4_{2} - #4&\matel{#1}{2}{3}&\ldots& \matel{#1}{2}{#3 -1}& \matel{#1}{2}{#3}\cr
0& 0&#4_{3} - #4&\ldots& \matel{#1}{3}{#3 - 1}& \matel{#1}{3}{#3}\cr
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr
0& 0&0&\ldots&#4_{#2 - 1} - #4&\matel{#1}{#2 - 1}{#3}\cr
0& 0&0&\ldots& 0 &#4_{#3}- #4\cr}}
%
%
\def\onecolmat#1#2#3{%
\pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#3}\cr
0& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#3}\cr
\vdots&\vdots&\ddots&\vdots\cr
0& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#3}\cr}}
%

\def\idmatrix{%
\pmatrix{\matel{1}{}{}& \matel{0}{}{}&\ldots& \matel{0}{}{}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{0}{}{}& \matel{0}{}{}&\ldots& \matel{1}{}{}\cr}}
%
\def\diagmat#1#2{%
\pmatrix{#1_{1}&        &\ldots& \cr
                & #1_{2}  &\ldots& \cr
\vdots&\vdots&\ddots&\vdots\cr
      &      &\ldots& #1_{#2}\cr}}
%
\def\expdiagmat#1#2{%
\pmatrix{e^{i #1_{1}}&        &\ldots& \cr
                & e^{i #1_{2}}  &\ldots& \cr
\vdots&\vdots&\ddots&\vdots\cr
      &      &\ldots& e^{i #1_{#2}}\cr}}
%

%
\def\diagmatso#1#2#3#4{%
\pmatrix{
I_{#3}&         &         & \ldots &         \cr
      & -I_{#4} &         &        &         \cr
      &         & #1_{1}  & \ldots &         \cr
\vdots&         &\vdots   & \ddots & \vdots  \cr
      &         &         & \ldots & #1_{#2} \cr}
}
%

%
\def\diagmatsob#1#2#3#4{%
\pmatrix{
 #1_{1}  & \ldots &        &        &        \cr
\vdots   & \ddots & \vdots &        &\vdots  \cr
         & \ldots & #1_{#2}&        &        \cr
         &        &        &-I_{#4} &        \cr
\ldots   &        &        &        & I_{#3} \cr
}
}
%

%
\def\diagmatsobb#1#2#3{%
\pmatrix{
 #1_{1}  & \ldots &        &        \cr
\vdots   & \ddots & \vdots &        \cr
         & \ldots & #1_{#2}&        \cr
\ldots   &        &        &  I_{#3}\cr
}
}
%

%
\def\diagmatlsobb#1#2#3{%
\pmatrix{
 #1_{1}  & \ldots &        &        \cr
\vdots   & \ddots & \vdots &        \cr
         & \ldots & #1_{#2}&        \cr
\ldots   &        &        &  0_{#3}\cr
}
}
%

\def\diagmatrecm#1#2{%
\pmatrix{#1_{1}&        &\ldots& \cr
                & #1_{2}  &\ldots& \cr
\vdots&\vdots&\ddots&\vdots\cr
      &      &\ldots& #1_{#2}\cr
 0    &  \vdots & \ldots & 0\cr
\vdots&\vdots&\ddots&\vdots\cr
 0    &  \vdots & \ldots & 0\cr
}}
%

\def\diagmatrecn#1#2{%
\pmatrix{#1_{1}&        &\ldots& & 0 &\ldots &0\cr
                & #1_{2}  &\ldots& & 0&\ldots& 0 \cr
\vdots&\vdots&\ddots&\vdots& 0 &\vdots & 0\cr
      &      &\ldots& #1_{#2}& 0 & \ldots & 0\cr}}
%

%
\def\kappamatrix{%
\pmatrix{
0        & \kappa_1&        &              &              \cr
-\kappa_1&     0   &\kappa_2&              &              \cr
         & -\kappa_2& 0     &\ddots        &              \cr
         &          &\ddots &\ddots        &\kappa_{n-1}  \cr
         &          &       &-\kappa_{n-1} & 0            \cr}}
%

%
\def\omegamatrix#1#2{%
\pmatrix{
0        & #1_{1\, 2}&        &              &              \cr
-#1_{1\, 2}&     0   &#1_{2\, 3}&              &              \cr
         & -#1_{2\, 3}& 0     &\ddots        &              \cr
         &          &\ddots &\ddots        &#1_{#2-1\, #2}  \cr
         &          &       &-#1_{#2-1\, #2} & 0            \cr}}
%


\def\gendet#1#2{%
\dmatrice{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#2}\cr
\matel{#1}{2}{1}& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#2}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}\cr}}
%
%
\def\charpoly#1#2#3{%
\dmatrice{\matel{#1}{1}{1} - #3& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#2}\cr
\matel{#1}{2}{1}& \matel{#1}{2}{2} - #3&\ldots& \matel{#1}{2}{#2}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}- #3\cr}}
%
%
\def\vecdet#1#2#3{%
\dmatrice{\matel{#1}{1}{1}& \ldots& \matel{#1}{1}{#2 -1}& #3_{1}\cr
\matel{#1}{2}{1}& \ldots&\matel{#1}{2}{#2 -1}& #3_{2}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1}&\ldots& \matel{#1}{#2}{#2 -1}& #3_{#2}\cr}}
%
%
\def\detb#1{%
\dmatrice{\matel{#1}{1}{1}& \matel{#1}{1}{2}& \matel{#1}{1}{3}\cr
\matel{#1}{2}{1}& \matel{#1}{2}{2}& \matel{#1}{2}{3}\cr
\matel{#1}{3}{1}& \matel{#1}{3}{2}& \matel{#1}{3}{3}\cr}}
%
%
\def\detc#1#2#3#4#5#6#7#8#9{%
\dmatrice{#1& #2& #3\cr
#4& #5& #6\cr
#7& #8& #9\cr}}
%
\newcommand\ndetc[9]{%
\begin{tabular}{|ccc|}
$#1$ & $#2$ & $#3$\\
$#4$ & $#5$ & $#6$\\
$#7$ & $#8$ & $#9$
\end{tabular}
}


%
\def\detcb#1#2#3#4#5#6#7#8#9{%
\dmatriceb{#1& #2& #3\cr
#4& #5& #6\cr
#7& #8& #9\cr}}
%
%
%
\def\deta#1#2#3#4{%
\dmatrice{#1& #2\cr
#3& #4\cr}}


\newcommand\ndeta[4]{%
\begin{tabular}{|cc|}
$#1$ & $#2$\\
$#3$ & $#4$
\end{tabular}
}


\def\resultant#1#2#3#4{%
\dmatrice{
#1_0 & #1_1 & \cdots & \cdots & #1_{#3} & 0 &\cdots &\cdots &\cdots &\cdots & 0\cr
0 &  #1_0 & #1_1 & \cdots & \cdots & #1_{#3} & 0 &\cdots &\cdots &\cdots  & 0\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
0 & \cdots &\cdots  &\cdots &\cdots & 0 & #1_0 & #1_1 & \cdots & \cdots &  #1_{#3}\cr
#2_0 & #2_1 & \cdots &  \cdots &\cdots &\cdots &\cdots & #2_{#4} & 0 & \cdots & 0\cr
0 &  #2_0 & #2_1 & \cdots &  \cdots & \cdots &\cdots &\cdots & #2_{#4} &  0 &\cdots\cr
\cdots  &\cdots  &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr
0 & \cdots & 0 & #2_0 & #2_1 &  \cdots & \cdots&\cdots &\cdots &\cdots   & #2_{#4}\cr
}}

\def\resultantht#1#2#3#4#5{%
\dmatrice{
#1_0 & #5#1_1 & \cdots & \cdots & #5^{#3}#1_{#3} & 0 &\cdots &\cdots &\cdots &\cdots & 0\cr
0 &  #1_0 & #5#1_1 & \cdots & \cdots & #5^{#3}#1_{#3} & 0 &\cdots &\cdots &\cdots  & 0\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
0 & \cdots &\cdots  &\cdots &\cdots & 0 & #1_0 & #5#1_1 & \cdots & \cdots &  #5^{#3}#1_{#3}\cr
#2_0 & #5#2_1 & \cdots &  \cdots &\cdots &\cdots &\cdots & #5^{#4}#2_{#4} & 0 & \cdots & 0\cr
0 &  #2_0 & #5#2_1 & \cdots &  \cdots & \cdots &\cdots &\cdots & #5^{#4}#2_{#4} &  0 &\cdots\cr
\cdots  &\cdots  &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr
0 & \cdots & 0 & #2_0 & #5#2_1 &  \cdots & \cdots&\cdots &\cdots &\cdots   & #5^{#4}#2_{#4}\cr
}}

\def\resultanthtt#1#2#3#4#5{%
\dmatrice{
#5#1_0 & #5^{2}#1_1 & \cdots & \cdots & #5^{#3+1}#1_{#3} & 0 
&\cdots &\cdots &\cdots &\cdots & 0\cr
0 &  #5^{2}#1_0 & #5^{3}#1_1 & \cdots & \cdots & #5^{#3+2}#1_{#3} & 0 
&\cdots &\cdots &\cdots  & 0\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
\cdots  &\cdots  &\cdots  & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr
0 & \cdots &\cdots  &\cdots &\cdots & 
0 & #5^{#4}#1_0 & #5^{#4+1}#1_1 & \cdots & \cdots &  #5^{#3+#4}#1_{#3}\cr
#5#2_0 & #5^{2}#2_1 & \cdots &  \cdots &\cdots &\cdots &\cdots &
 #5^{#4+1}#2_{#4} & 0 & \cdots & 0\cr
0 &  #5^{2}#2_0 & #5^{3}#2_1 & \cdots &  \cdots & \cdots &\cdots &\cdots &
 #5^{#4+2}#2_{#4} &  0 &\cdots\cr
\cdots  &\cdots  &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr
0 & \cdots & 0 & #5^{#3}#2_0 & #5^{#3+1}#2_1 &  
\cdots & \cdots&\cdots &\cdots &\cdots   & #5^{#3+#4}#2_{#4}\cr
}}

\def\resultantb#1#2#3#4{%
\dmatrice{
1 & #1_1 & \cdots & \cdots & \cdots & #1_{#3} & 0 &  \cdots &\cdots & 0\cr
0 &  1 & #1_1 & \cdots &  \cdots &  \cdots & #1_{#3} & 0 & \cdots&\cdots\cr
\cdots  &\cdots  &\cdots &\cdots & \cdots & \cdots &  \cdots & \cdots &\cdots &\cdots \cr
0 & \cdots & 0 & 1 & #1_1 & \cdots & \cdots&\cdots&\cdots& #1_{#3}\cr
#3 & (#3-1)#2_1 & \cdots & \cdots &#2_{#4} & 0 &  \cdots &  \cdots &  \cdots & 0\cr
0 &  #3 & (#3-1)#2_1 & \cdots &\cdots& #2_{#4} & 0 & \cdots &\cdots &  \cdots & \cr
\cdots  &\cdots  &\cdots &\cdots & \cdots &\cdots & \cdots &  \cdots & \cdots &\cdots \cr
0 & \cdots & \cdots & 0 & #3 & (#3-1)#2_1 &  \cdots &  \cdots & \cdots& #2_{#4}\cr
}}

%
\def\vandermonde#1#2{%
\dmatrice{1& 1&\ldots& 1\cr
#1_1& #1_2&\ldots& #1_{#2}\cr
#1_{1}^{2}& #1_{2}^{2}&\ldots& #1_{#2}^{2}\cr
\vdots&\vdots&\ddots&\vdots\cr
#1_{1}^{#2 - 1}& #1_{2}^{#2 - 1}&\ldots& #1_{#2}^{#2 - 1}\cr}}
%
%
\def\svandermonde#1#2{%
\dmatrice{1& 1&\ldots& 1\cr
0& #1_2 - #1_1&\ldots& #1_{#2} - #1_1\cr
0& #1_{2}(#1_{2} - #1_1)&\ldots& #1_{#2}(#1_{#2} - #1_1)\cr
\vdots&\vdots&\ddots&\vdots\cr
0& #1_{2}^{#2 - 2}(#1_{2} - #1_1)&\ldots& #1_{#2}^{#2 - 2}(#1_{#2} - #1_1)\cr}}
%

\def\sindeta#1#2{%
\dmatrice{
\sin #2_1 & \sin #2_2 & \ldots & \sin #2_{#1}\cr
-\sin^3 #2_1 & -\sin^3 #2_2 & \ldots & -\sin^3 #2_{#1}\cr
\vdots&\vdots&\ddots&\vdots\cr
(-1)^{#1 -1}\sin^{2#1-1} #2_1 & (-1)^{#1 -1}\sin^{2#1-1} #2_2 & \ldots & 
(-1)^{#1 -1}\sin^{2#1-1} #2_{#1}\cr
}}

\def\sindetb#1#2{%
\dmatrice{
\sin #2_1 & \sin #2_2 & \ldots & \sin #2_{#1}\cr
\sin 2#2_1 & \sin 2#2_2 & \ldots & \sin 2#2_{#1}\cr
\vdots&\vdots&\ddots&\vdots\cr
\sin #1#2_1 & \sin #1#2_2 & \ldots & \sin #1#2_{#1}\cr
}}

\def\sindetc#1#2{%
\dmatrice{
 #2_1 &  #2_2 & \ldots &  #2_{#1}\cr
- #2_1^3 &  -#2_2^3 & \ldots &  -#2_{#1}^3\cr
\vdots&\vdots&\ddots&\vdots\cr
(-1)^{#1-1} #2_1^{2#1-1} & (-1)^{#1-1} #2_2^{2#1-1} & \ldots & 
(-1)^{#1-1} #2_{#1} ^{2#1-1}\cr
}}

%
\def\castel#1#2{%
\dmatrice{
1& \sigma_1(#1_{1},\ldots,#1_{#2})&\ldots&
\sigma_{#2}(#1_{1},\ldots,#1_{#2}) \cr
1& \sigma_1(#1_{2},\ldots,#1_{#2+1})&\ldots&
\sigma_{#2}(#1_{2},\ldots,#1_{#2+1}) \cr
1& \sigma_1(#1_{3},\ldots,#1_{#2+2})&\ldots&
\sigma_{#2}(#1_{3},\ldots,#1_{#2+2}) \cr
\vdots&\vdots&\ddots&\vdots\cr
1& \sigma_1(#1_{#2+1},\ldots,#1_{2#2})&\ldots&
\sigma_{#2}(#1_{#2+1},\ldots,#1_{2#2}) \cr}}
%

%
%\def\natnums{{\bf N}}
\def\natnums{{\mathbb N}}
%\def\integs{{\bf Z}}
\def\integs{{\mathbb Z}}
%\def\rats{{\bf Q}}
\def\rats{{\mathbb Q}}
%\def\reals{{\bf R}}
\def\reals{{\mathbb R}}
%\def\complex{{\bf C}}
\def\complex{{\mathbb C}}
\def\Ker{{\rm Ker}\,}
\def\coker{{\rm Coker}\,}
\def\Kerof#1{\Ker(#1)}
\let\Immag=\Im
\def\Im{{\rm Im}\,}
\def\Imof#1{\Im(#1)}
\def\dual#1{#1^{*}}
\def\dualh#1{{#1}'}
\def\bdual#1{#1^{**}}
\def\id{{\rm id}}
\def\dimm{{\rm dim}}
\def\codim{{\rm codim}}
\def\rg{{\rm rk}}
\def\card{{\rm card}}
\def\deg#1{{\rm deg}#1}
\def\mdeg{m}
\def\ndeg{n}
\def\pdeg{p}
\def\qdeg{q}
\def\ddeg{d}
\def\Ndeg{N}
\def\Jet{{\rm Jet}}
\def\coJet{\hbox{co-Jet}}
\def\gr{{\rm gr}}
\def\ratio{{\rm ratio}}
\def\ideal#1{\mfrac{#1}}
\def\hatplus{\>\widehat{+}\>}
\def\hatminus{\>\widehat{-}\>}
\def\pcompl#1{\widetilde{#1}}
\def\ptinf#1{{#1}_{\infty}}
%\def\ptinf#1{\vector{#1}_{\infty}}
\def\homog#1{#1_{*}}
\def\tensalg{{\rm T}}
\def\salg{{\rm S}}
\def\symalg{{\rm Sym}}
\def\extalg{\bigwedge}
\def\domm{{\rm dom}}
\def\Bezierbc#1#2#3#4#5{{\cal B}\Big[#1_{#2},\ldots,#1_{#3};\, [#4,\,#5]\Big]}
\def\Bezierc#1#2#3{{\cal B}\Big[#1;\, [#2,\,#3]\Big]}
%\def\prospac#1#2{{\bf P}^{#1}_{#2}}
\def\prospac#1#2{{\mathbb{P}}^{#1}_{#2}}
%\def\rprospac#1{{\bf RP}^{#1}}
\def\rprospac#1{{\mathbb{RP}}^{#1}}
%\def\cprospac#1{{\bf CP}^{#1}}
\def\cprospac#1{{\mathbb{CP}}^{#1}}
\def\projs#1{{\bf P}(#1)}
\def\Proj{\mathrm{Proj}}
\def\bprojs#1{{\bf P}\bigl(#1\bigr)}
%\def\projr#1{{\bf P}^{#1}}
\def\projr#1{{\mathbb{P}}^{#1}}
\def\projv{\mathrm{Proj}}
\def\affs{\s{E}}
\def\christofa#1#2#3{[#1\, #2;\, #3]}
\def\christofb#1#2#3{\Gamma_{#1\, #2}^{#3}}
%\def\pairtb#1#2{\left< #1,\; #2\right>}
\def\pairtb#1#2{\left< #1, #2\right>}
\def\mbold#1{{\bf #1}}
%\def\mfrac#1{{\EuFrak #1}}
\def\mfrac#1{{\mathfrak{#1}}}
\def\voldeta#1#2{%
\dmatrice{
\matel{#1}{0}{1}&\matel{#1}{0}{2}&\ldots & \matel{#1}{0}{#2}& 1\cr
\matel{#1}{1}{1}&\matel{#1}{1}{2}&\ldots & \matel{#1}{1}{#2}& 1\cr
\vdots&\vdots&\ddots&\vdots&\vdots\cr
\matel{#1}{#2}{1}&\matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}& 1\cr}}
%

\def\voldetb#1#2{%
\dmatrice{
\matel{#1}{1}{1} - \matel{#1}{0}{1}&\matel{#1}{1}{2} - \matel{#1}{0}{2}&
\ldots & \matel{#1}{1}{#2} - \matel{#1}{0}{#2}\cr
\matel{#1}{2}{1} - \matel{#1}{0}{1}&\matel{#1}{2}{2} - \matel{#1}{0}{2}&
\ldots & \matel{#1}{2}{#2} - \matel{#1}{0}{#2}\cr
\vdots&\vdots&\ddots&\vdots\cr
\matel{#1}{#2}{1} - \matel{#1}{0}{1}&\matel{#1}{#2}{2} - \matel{#1}{0}{2}&
\ldots&\matel{#1}{#2}{#2} - \matel{#1}{0}{#2}\cr}}
%

\def\gramdet#1#2{%
\dmatrice{
\norme{#1_1}^2 &\pairt{#1_1}{#1_2}&\ldots & \pairt{#1_1}{#1_{#2}}\cr
\pairt{#1_2}{#1_1} &\norme{#1_2}^2&\ldots & \pairt{#1_2}{#1_{#2}}\cr
\vdots&\vdots&\ddots&\vdots\cr
\pairt{#1_{#2}}{#1_1} &\pairt{#1_{#2}}{#1_2}&\ldots & \norme{#1_{#2}}^2\cr
}}
%

\def\chull#1{\s{C}(#1)}
